Problem 33 - Precalculus: A Concise Course

Question

Use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions. \(\cos \theta=\frac{1}{3}\) (a) \(\sin \theta\) (b) \(\tan \theta\) (c) \(\sec \theta\) (d) \(\csc \left(90^{\circ}-\theta\right)\)

Step-by-Step Solution

Answer

The values are: (a) \(\sin \theta = \frac{2\sqrt{2}}{3}\), (b) \(\tan \theta = 2\sqrt{2}\), (c) \(\sec \theta = 3\), (d) \(\csc \left(90^{\circ}-\theta\right) = 3\).

Step 1: Calculate \(\sin \theta\)

We have \(\cos \theta = \frac{1}{3}\). Using the identity \(\sin^{2}\theta + \cos^{2}\theta = 1\), we can find \(\sin \theta = \sqrt{1-\cos^{2}\theta} = \sqrt{1-\left(\frac{1}{3}\right)^{2}} = \sqrt{1-\frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}\).

Step 2: Calculate \(\tan \theta\)

\(\tan \theta\) is defined as \(\frac{\sin \theta}{\cos \theta}\). Hence, \(\tan \theta = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} = 2\sqrt{2}\).

Step 3: Calculate \(\sec \theta\)

\(\sec \theta\) is the reciprocal of \(\cos \theta\). Hence, \(\sec \theta = \frac{1}{ \cos \theta} = \frac{1}{\frac{1}{3}} = 3\).

Step 4: Calculate \(\csc \left(90^{\circ}-\theta\right)\)

The cosecant function is the reciprocal of the sine function. However, by the cofunction identity \(\csc \left(90^{\circ}-\theta\right) = \sec \theta\). Therefore, \(\csc \left(90^{\circ}-\theta\right) = \sec \theta = 3\).

Key Concepts

Sine Function

The sine function, denoted as \( \sin \theta \), is a fundamental trigonometric function. It represents the y-coordinate of a point on the unit circle — that is, a circle with a radius of one. In the unit circle, each point can be described using \( \sin \) and \( \cos \):

• \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• In terms of a right-angled triangle, \( \sin \theta \) is the ratio of the length of the opposite side to the hypotenuse for an angle \( \theta \).

When given \( \cos \theta = \frac{1}{3} \), we can use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \sin \theta \). Substitute \( \cos \theta \) into the identity to solve for \( \sin^2 \theta \), which simplifies to \( \sin \theta = \frac{2 \sqrt{2}}{3} \). This calculation demonstrates using the Pythagorean identity to evaluate the sine function when directly dealing with another trigonometric value.

Tangent Function

The tangent function, \( \tan \theta \), is another key trigonometric function. It is defined as the ratio between the sine and cosine of the same angle, \( \theta \).

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• \( \tan \theta \) can also be perceived as the ratio of the length of the opposite side to the adjacent side in a right-angled triangle.

With \( \cos \theta = \frac{1}{3} \) and \( \sin \theta = \frac{2\sqrt{2}}{3} \), plug these values into the formula for tangent. This yields \( \tan \theta = \frac{2\sqrt{2}}{3} \div \frac{1}{3} = 2\sqrt{2} \). This illustrates how the tangent function synthesizes the other two primary trigonometric functions to provide a different method of angle representation.

Trigonometric Functions

Trigonometric functions are critical mathematical tools used to relate angles to side lengths in triangles, while broadly applying to various fields such as physics and engineering.
These functions include:

• \( \sin \): Sine
• \( \cos \): Cosine
• \( \tan \): Tangent
• \( \sec \): Secant
• \( \csc \): Cosecant
• \( \cot \): Cotangent

Each function defines a unique ratio of sides in a right triangle, providing distinct perspectives on how angles relate to side lengths. For example, \( \sec \theta \), the secant, is the reciprocal of \( \cos \theta \). In our example, \( \sec \theta = 3 \), arising from \( \frac{1}{\cos \theta} \). Understanding these relationships allows students to tackle complex problems across contexts, reinforcing the independent utility of each function.

Cofunction Identities

Cofunction identities are a specialized set of trigonometric identities expressing the notion that certain trigonometric functions of complementary angles are equal. These come into play, particularly when angles sum to \(90^\circ \).
For example:

• \( \sin \theta = \cos(90^\circ - \theta) \)
• \( \tan \theta = \cot(90^\circ - \theta) \)
• \( \sec \theta = \csc(90^\circ - \theta) \)

Focusing on \( \csc(90^\circ - \theta) \), it translates to \( \sec \theta \), demonstrating cofunctionality. In the given problem, since \( \sec \theta = 3 \), \( \csc(90^\circ - \theta) \) also becomes 3. Cohesiveness within these identities simplifies calculating trigonometric values, especially when only one angle measurement is initially available.